First of all, I'm sorry if this isn't the kind of question that should be made in MathOverflow. I read the FAQ and I didn't consider this (that) inappropriate. I couldn't resist! People here are experts. All the experience here could help me.

I'm starting my undergraduate studies in september. I was studying computer science but I'm switching majors. Now I'm going to study mathematics. I don't want to fail any class, and some recommendation from you would be useful.

So I'd like to ask two questions:

What would have been good for you to know before you started studying mathematics that you didn't know?

Please explain why you want to graduate as fast as you can. Unless there are insurmountable financial reasons, it sounds like an unwise idea: you should enjoy your time as an undergraduate, and have as much opportunity as possible to explore potential interests (math and otherwise) so that you are well-equipped as you can be for whatever you do after college. Your professors in college can surely give you useful advice on the things you ask about.
–
BoyarskyJun 18 '10 at 23:18

Actually, I don't want to graduate as fast as I can, but I don't want to fail any course. That's what I meant by that. I feel like I lost time studying computer science and I don't want to waste any more.
–
werrketJun 18 '10 at 23:26

This question is asking a great many different things, all at once. I'd strongly encourage rewriting it (before any answers appear!) to be more focused. Perhaps you can postpone some pieces for later?
–
Scott Morrison♦Jun 19 '10 at 0:39

2

That's a good start, cutting down to just two things!
–
Scott Morrison♦Jun 19 '10 at 4:26

4 Answers
4

If you get stuck on a concept, don't read only one book to try and learn it. Look for other books. Sometimes, some of the most subtle changes in presentation can help give enough perspective to get the crucial insight required to really grok a concept. The internet is awesome nowadays; alternative reading material is now easily obtained.

I had of my most influential insights from textbooks written in the 1970s (my undergrad days were in the 1990s), from the university library, which I decided to read out of curiosity when I procrastinated there. (Needless to say, my choice to procrastinate at the university library was made before the advent of widespread reading/writing of blogs as a means of procrastination.)

You need to know that you know nothing.
Seriously though, that should be the guiding thought when learning mathematics. There is always some subtlety that can be missed or a concept that can be understood the wrong way (and is by many people) when you are too sure of your knowledge.

The answers surely depend on what kind of mathematics course, and where (you give no clues at all). But here's one traditional point: make sure you are absolutely solid on the difference between necessary, sufficient and necessary-and-sufficient conditions. (Millions are not!) This has nothing much to do with the algorithmic and semi-algorithmic aspects of calculus, but has everything to do with the "comfort zone" one talks about for "being a good mathematician".

After that, try finding the fallacy in the argument that a transitive, symmetric relation must be an equivalence relation.

You're right; I forgot to tell where I'm going to be studying. I live in Venezuela and I'm going to study in the Simon Bolívar University. This is the curriculum: mat.coord.usb.ve/pensum_eymc.htm. It's in spanish but I don't think it's very difficult to understand if you want to check it out.
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werrketJun 19 '10 at 14:18

1

Plenty of analysis. Starting from a computer science background, you should probably concentrate on developing some feeling for inequalities.
–
Charles MatthewsJun 19 '10 at 16:33

Think-Rethink, Do-Redo, Check-Recheck! Then discuss the solution to someone else, but not necessarily.

This is the algorithm to find the answer to any mathematical problem.

Also read the many pieces of advice you can get from mathematicians you consider good, up to your taste. Wikipedia articles and histories of mathematicians have always been a source of delight and instruction in my not-so-yet-long mathematical endeavors.